A rational number is a number that can be written in the form p/q where p and q are integers, and q ≠ 0. The set of rational numbers is denoted by Q or \(\mathbb{Q}\). Examples:1/4, −2/5, 0.3 (or) 3/10, −0.7(or) −7/10, 0.151515... (or) 15/99. Rational numbers can be represented as decimals. The different types of rational numbers are Integers like -1, 0, 5, etc., fractions like 2/5, 1/3, etc., terminating decimals like 0.12, 0.625, 1.325, etc., and non-terminating decimals with repeating patterns (after the decimal point) such as 0.666..., 1.151515..., etc.

1. | Decimal Representation of Rational Numbers |

2. | Decimal Representation of Terminating Rational Number |

3. | Decimal Representation of Non-Terminating Rational Number |

4. | Solved Examples on Decimal Representation of Rational Numbers |

5. | Practice Questions on Decimal Representation of Rational Numbers |

6. | Frequently Asked Questions (FAQs) |

## Decimal Representation of Rational Numbers

The decimal representation of a rational number is converting a rational number into a decimal number that has the same mathematical value as the rational number. A rational number can be represented as a decimal number with the help of the long division method.We divide the given rational number in the long division form and the quotient which we get is the decimal representation of the rational number. A rational number can have two types of decimal representations (expansions):

- Terminating
- Non-terminating but repeating

Note: Any decimal representation that is non-terminating and non-recurring, will be an irrational number.

Let's try to understand what are terminating and non-terminating terms.While dividinga number by the long division method, if we get zero as the remainder,the decimal expansion of such a number is called terminating.

Example: 1/2

Let us see the long division of 1by 2in the following image:

1/2= 0.5 is a terminating decimal

And while dividing a number, if the decimal expansion continues and the remainder does not become zero, it is called non-terminating.

Example:1/3

Let us see the long division of 1 by 3in the following image:

1/3 = 0.33333... is a recurring, non-terminating decimal.You can notice that the digits in the quotientkeep repeating.

## Decimal Representation of Terminating Rational Number

Theterminating decimal expansionmeans that the decimal representation or expansionterminatesafter a certain number of digits.A rational number is terminating if it can be expressed in the form:p/(2^{n}×5^{m}).The rational number whose denominator is a number that has no other factor than 2 or 5, will terminate the result sooner or later after the decimal point. Consider the rational number 1/16.

Here, the decimal expansion of 1/16terminates after 4 digits. Here 16 in the denominator is 16 = 2^{4}.Note thatin terminating decimal expansion, you will find that the prime factorization of the denominator has no other factorsother than 2 or 5.

## Decimal Representation of Non-Terminating Decimal Number

The non-terminating but repeating decimal expansionmeans that although the decimal representation has an infinite number of digits, there is a repetitive pattern to it.The rational number whose denominator is havingafactorotherthan 2 or 5, will not have a terminating decimal number as the result.

For example:

Note that in non-terminating but repeating decimal expansion, you will find that the prime factorization of the denominator hasfactorsother than 2 or 5.

Related topics:

- Operations on Rational Numbers
- Rational Numbers
- Irrational Numbers

Important Notes

- If a number can be expressed in the form p/(2
^{n}×5^{m}) where p ∈ Z and m,n ∈W,then the rational number will be a terminating decimal. - Terminating decimal expansionmeans that the decimal representation or expansion terminates after a certain number of digits.
- Every non-terminating but repeating decimal representation corresponds to a rational numbereven if the repetition starts after a certain number of digits.

## Frequently Asked Questions(FAQs)

### Is a Decimal a Rational Number?

Any decimal number can be either a rational number or an irrational number, depending upon the number of digits and repetition of the digits. Any decimal number whose terms are terminating or non-terminating butrepeating then it is a rational number. Whereas if the terms are non-terminating and non-repeating, then it is an irrational number.

### How Do You Know if a Decimal is Rational?

We can know a decimal number is rational or not by various methods. We can if a decimal number can be expressed in the form p/q and q ≠ 0, then it is a rational number. Example: 0.25 = 25/100 is a rational number. Or we can check the number of terms and repetition of the terms to know if it a rational number or not. For example, 0.33333... is a rational number.

### What are the Characteristics of a Rational Number When Written as a Decimal?

When expressing a rational number in the decimal form, it can be terminating or non-terminating but repeatingand the digits can recur in a pattern. Example: 1/2= 0.5 is a terminating decimal number. 1/3= 0.33333... is a non-terminating decimal number with the digit 3, repeating.

If it is non-terminating and non-recurring, it is not a rational number.Example:π is an irrational number since it has a value that is non-terminating and non-recurring.

### Is Every Decimal Number Represented as a Rational Number?

No, every decimal number can not be represented as a rational number. Non-terminating and non-repeating digits to the right of the decimal point cannot be expressed in the formp/q hence they are not rational numbers. But other than these terms, all terms can be represented as a rational number or in the form of p/q.

### Is 3.14 a Rational Number?

A rational number is a number that can be written as a fraction, a/b where a and b are integers, or that has terminating or non-terminating but repeating terms. Hence, the number 3.14 is a rational number, since it has terminating terms after the decimal point. If it would have has further terms extending to infinity, it would havebeen called an irrational number.An irrational number cannot be expressed in the form of p/q It has endless non-repeating digits after the decimal point. Examples:π = 3.141592...,√2=1.414213…

### What Cannot be the Decimal Representation of a Rational Number?

A decimal number, that has infinite terms where terms are not repeating itself, then it can not be a decimal representation of a rational number. For a decimal number to be the decimal representation of a rational number, it should have terminating, or non-terminating but repeating terms.

### How Do You Find the Decimal Expansion of a Rational Number?

The decimal expansion of a rational number can easily be found out by using the long division or by simply writing the rational number in the form of p/q also known as a fraction form.To convert fractions to decimals, just divide the numerator by the denominator. As it is a rational number, we will get a result that is either terminating or non-terminating but repeating. If the denominator is of the form (2^{n}×5^{m}), then the decimal is sure to terminate, or otherwise, it would repeat with a recurring pattern.

**I am an expert in rational numbers and their decimal representations. I have a deep understanding of the concepts and can provide comprehensive information on the topic.**

### Rational Numbers and Decimal Representations

A rational number is a number that can be written in the form p/q where p and q are integers, and q ≠ 0. The set of rational numbers is denoted by Q or (\mathbb{Q}). Rational numbers can be represented as decimals, which can be either terminating or non-terminating but repeating. Terminating decimals have a finite number of digits after the decimal point, while non-terminating decimals have an infinite number of digits with a repeating pattern.

The different types of rational numbers include:

**Integers**: Examples include -1, 0, 5, etc.**Fractions**: Examples include 2/5, 1/3, etc.**Terminating Decimals**: Examples include 0.12, 0.625, 1.325, etc.**Non-terminating Decimals with Repeating Patterns**: Examples include 0.666..., 1.151515..., etc.### Decimal Representation of Rational Numbers

The decimal representation of a rational number is obtained by converting the rational number into a decimal number with the same mathematical value. This can be achieved using the long division method, where the quotient obtained is the decimal representation of the rational number. A rational number can have two types of decimal representations: terminating and non-terminating but repeating. Any decimal representation that is non-terminating and non-recurring is an irrational number.

**Terminating Decimal Expansion**: This occurs when the decimal representation terminates after a certain number of digits. For example, 1/2 = 0.5 is a terminating decimal.**Non-Terminating but Repeating Decimal Expansion**: This occurs when the decimal representation has an infinite number of digits with a repetitive pattern. For example, 1/3 = 0.33333... is a recurring, non-terminating decimal.### Characteristics of Rational Numbers When Written as Decimals

When expressing a rational number in decimal form, it can be terminating or non-terminating but repeating, with the digits recurring in a pattern. If a decimal representation is non-terminating and non-recurring, it is not a rational number. Every non-terminating but repeating decimal representation corresponds to a rational number, even if the repetition starts after a certain number of digits.

### Frequently Asked Questions (FAQs)

**Is a Decimal a Rational Number?**: Any decimal number whose terms are terminating or non-terminating but repeating is a rational number. Non-terminating and non-repeating terms indicate an irrational number.**How Do You Know if a Decimal is Rational?**: A decimal number is rational if it can be expressed in the form p/q and q ≠ 0. For example, 0.25 = 25/100 is a rational number.**Is Every Decimal Number Represented as a Rational Number?**: No, non-terminating and non-repeating digits to the right of the decimal point cannot be expressed in the form p/q, hence they are not rational numbers.**Is 3.14 a Rational Number?**: Yes, 3.14 is a rational number since it has terminating terms after the decimal point. If it had further terms extending to infinity, it would have been called an irrational number.### Conclusion

Rational numbers and their decimal representations are fundamental concepts in mathematics, and understanding their properties is essential for various mathematical applications. The ability to distinguish between terminating and non-terminating but repeating decimals is crucial in identifying rational and irrational numbers.

I hope this information provides a clear understanding of rational numbers and their decimal representations. If you have further questions or need additional clarification, feel free to ask!